## Abstract

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if S: (a_{n}) _{n}εℤ→ a_{n-1n}εℤ is the shift operator acting on the weighted space of sequences ℓ^{2}ℤ , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if lim^{n}→ +rm log}ω(-n){n}=0} . On the other hand one can see that S is not cyclic if the series ∑_{n}geq 1} logω (-n)}/n^{2} diverges. We show that the question of Herrero whether either S or S* is cyclic on ℓω^{2} ℤ admits a positive answer when the series ∑_{n}εℤ rm log} ^{S}{n}||/(n^{2}+1)} is convergent. We also prove completeness results for translates in certain Banach spaces of functions on ℝ.

Original language | English |
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Pages (from-to) | 293-322 |

Number of pages | 30 |

Journal | Mathematische Annalen |

Volume | 341 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2008 |